BALTIC-WAY 2025 Problem 11

Let $A_1, \ldots, A_n$ be a sequence of points on the Euclidean plane. Some of the points may coincide, but the sequence contains at least two distinct points. A Baltic Way is a route defined by a permutation $B_1, \ldots, B_n$ of the points $A_1, \ldots, A_n$. The length of a Baltic Way is

$$|B_1B_2| + |B_2B_3| + \cdots + |B_{n-1}B_n| + |B_nB_1|,$$

where $|PQ|$ denotes the Euclidean distance between points $P$ and $Q$.

Let $K$ and $L$ be the lengths of some two Baltic Ways on the same sequence of $n$ points. For a given $n$, determine the maximum possible value of $K/L$ over all possible sequences $A_1, \ldots, A_n$.